Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 613 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.What is the smallest number you must add to 354 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
2. 2.Which of these numbers is divisible by 3? (a) 240 (b) 238 (c) 245 (d) 241
3. 3.What is the units digit of 8^52? (a) 2 (b) 6 (c) 4 (d) 8
4. 4.What is the units digit of 4^12? (a) 4 (b) 7 (c) 6 (d) 5
5. 5.What is the units digit of 3^10? (a) 7 (b) 1 (c) 9 (d) 3
6. 6.What is the smallest number you must add to 238 to make it divisible by 3? (a) 1 (b) 2
7. 7.What is the remainder when 142 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
8. 8.Which of these numbers is divisible by 3? (a) 685 (b) 679 (c) 682 (d) 681
9. 9.What is the units digit of 8^57? (a) 4 (b) 6 (c) 8 (d) 2
10. 10.Which of these numbers is divisible by 6? (a) 1483 (b) 1488 (c) 1498 (d) 1489
11. 11.What is the remainder when 846 is divided by 11? (a) 9 (b) 1 (c) 8 (d) 10
12. 12.What is the units digit of 9^40? (a) 2 (b) 1 (c) 9 (d) 0
13. 13.What is the units digit of 3^23? (a) 7 (b) 9 (c) 3 (d) 1
14. 14.Which of these numbers is divisible by 6? (a) 3239 (b) 3245 (c) 3247 (d) 3240
15. 15.What is the smallest number you must add to 438 to make it divisible by 7? (a) 2 (b) 4 (c) 3 (d) 5
16. 16.What is the units digit of 2^22? (a) 6 (b) 4 (c) 8 (d) 2
17. 17.What is the smallest number you must add to 361 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1
18. 18.What is the smallest number you must add to 538 to make it divisible by 9? (a) 2 (b) 3 (c) 7 (d) 1
19. 19.What is the units digit of 7^41? (a) 7 (b) 9 (c) 3 (d) 1
20. 20.What is the remainder when 191 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 1
21. 21.What is the units digit of 4^40? (a) 4 (b) 5 (c) 7 (d) 6
22. 22.What is the units digit of 7^49? (a) 9 (b) 1 (c) 7 (d) 3
23. 23.What is the smallest number you must add to 961 to make it divisible by 12? (a) 1 (b) 10 (c) 9 (d) 11
24. 24.What is the units digit of 2^44? (a) 8 (b) 6 (c) 2 (d) 4
25. 25.What is the remainder when 354 is divided by 10? (a) 4 (b) 3 (c) 6 (d) 5
26. 26.What is the smallest number you must add to 735 to make it divisible by 10? (a) 6 (b) 5 (c) 7 (d) 4
27. 27.What is the remainder when 461 is divided by 11? (a) 1 (b) 9 (c) 10 (d) 8
28. 28.What is the smallest number you must add to 71 to make it divisible by 6? (a) 5 (b) 2 (c) 1 (d) 3
29. 29.Which of these numbers is divisible by 4? (a) 3584 (b) 3590 (c) 3586 (d) 3582
30. 30.What is the units digit of 4^23? (a) 5 (b) 6 (c) 4 (d) 3
31. 31.Which of these numbers is divisible by 8? (a) 2504 (b) 2498 (c) 2518 (d) 2505
32. 32.What is the units digit of 7^32? (a) 9 (b) 3 (c) 7 (d) 1
33. 33.Which of these numbers is divisible by 11? (a) 5469 (b) 5467 (c) 5482 (d) 5465
34. 34.What is the units digit of 4^37? (a) 3 (b) 4 (c) 6 (d) 5
35. 35.Which of these numbers is divisible by 11? (a) 5972 (b) 5973 (c) 5991 (d) 5975
36. 36.What is the units digit of 7^56? (a) 9 (b) 3 (c) 7 (d) 1
37. 37.Which of these numbers is divisible by 9? (a) 2350 (b) 2358 (c) 2364 (d) 2372
38. 38.What is the smallest number you must add to 226 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
39. 39.What is the remainder when 392 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
40. 40.What is the smallest number you must add to 228 to make it divisible by 10? (a) 1 (b) 2 (c) 3 (d) 8

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